Abstract:  The purpose of this paper is to show and discuss the differential equation used in analyzing the efficiency of a slider-bat versus a normal bat in an article.  The article comes to a conclusion on the efficiency depending on the batter desiring a shorter or longer swing time using the differential equation with extensive mathematical steps.

Introduction

    When a baseball player goes to choose his or her bat, what are they looking for?  A smart baseball player would get a bat that has a small moment of inertia during the swing, so that the bat can be swung quickly, and a large moment of inertia at collision for a maximum transfer of energy from the bat to the baseball.  In the baseball world there has been developments of baseball bats, which are considered to be "superbats"; the bats have a low moment of inertia during the swing time, and a high moment of inertia during the collision of the bat and the baseball.  How are these developments made?  A superbat is composed of a hollow bat with a sliding mass, which starts near the hands of the batter and ends up at the end of the bat at the collision time.  This allows for a concentration of mass at the hands when swinging, and a heavy end to transfer a large amount of energy at the end of the swing so that the ball will travel much farther than if hit with an ordinary bat.  In the article "Bat and Superbat", the effectiveness of the sliding mass system is analyzed with differential equations.  The swing time is first analyzed and then the collision time.  The author of the article, which performed the study is Herbert R. Bailey and was a Professor of Mathematics at Rose-Hulman Institute of Technology at the time of the article.  His research publications are in the fields of fluid flow, heat flow, vibration analysis, and diffusion.

Differential Equation

Swing

    The analyzing of the swing phase of the baseball bat must begin with the rotation equation, which describes the motion of a rigid body rotating around a fixed axis:

    In the above equation T is the net external torque,  the torque is the force times the distance from the fixed axis to the line of force, I  is the moment of inertia which is defined by the integral of r^2 dm, where dm is an element of mass which is at a distance r from the fixed axis, and finally Ö=d²O/dt², where O is the angle of rotation, and t is time. The torque during the swing can be called the Torque applied, Ta.  The assumption is made that the Ta by the batter can be expressed by the equation Ta=k*t, where k is a constant torque rate, and that the torque acts around the fixed vertical axis through the point at the end of the knob end of the bat, and it is also assumed that the bat is swung in a horizontal plane.  Due to the fact of the assumption that the torque is constant, makes it true that the torque increases during the swing at a constant linear rate, k, starting at a torque of zero and ending in at a maximum torque.  The article stated that an assumption of constant torque during the swing gave similar results as the assumption of non-constant torque, but the recorded values of the swing time and the batted-ball velocity were not as realistic.  The method used to analyze the problem was to analyze in terms of the motion of rigid bodies, which means separating the slider and bat into two rigid bodies, as shown in the Figure 1 below.  A Force Fs is exerted on the slider by the bat, and in return the slider exerts an equal but opposite force on the bat; it is important to see that Fs is perpendicular to the axis of the bat.  For the next equation to be developed, it must be noted that the net torque applied to the bat by the batter is Ta-Fs*rs, therefore the rotation equation T=I*Ö is then:

In equation (2) Ib is the moment of inertia of the bat about the vertical axis.

    Next the application of the translation equation F=ms*a to the motion of the slider, where the force and acceleration vectors are resolved into their radial components Fr and ar, and also their angular components Fo and ao. Due to the slider experiencing no friction Fr=0 and Fo=Fs.  Eventually within the article the translation equation is resolved into its radial and angular components, as follows:

After equations (3) and (4) are rewritten and combined the following equations are obtained for this modeling:
 
 

The next step is to solve the initial value problem with the initial conditions given:

In the above initial conditions, ro is the initial radial coordinate of the slider.  The next step was to let Oc represent the total swing angle, and solve the initial value problem with the initial conditions given in (7) .  By solving the initial value problem above it is determined at the time of the collision the values of time tc, angular velocity dOc/dt, slider position rsc and the slider velocity drsc/dO.  The values of dOc/dt and rsc are required to determine the velocity of the batted ball that will be determined in the collision phase.

In the article the author was not able to determine an explicit solution of the nonlinear system of equations (5) and (6), and the initial conditions in (7).  The author was able to solve the system numerically.

Collision

    In the last part of the article, the collision phase of the swing is analyzed to see the effects of a superbat and whether they are effective or not.  During the collision of the ball and the bat  the slider and bat will be considered one item with a mass msb=ms+mb.  The total moment of inertia of the slider and bat at the time of collision will be the sum of the moments of the bat, which is Ib, and the slider, which is ms*r²sc, which then becomes Isb=Ib+ms*r²sc, where rsc is the position of the slider at collision.  Next the article presents the integrated form of Newton's Second Law, F=ma, which is applied to the mass of the ball, ma, which is the integral from ti to tf of F(t) dt and that is equal to the net change in linear momentum, and is equal to:

F(t) is the force which the bat exerts on the ball, va*(ti) and va*(tf) are the velocities of the ball right before the impact and right after impact at ti and tf.  The article then presents a corresponding equation for the slider-bat, which is the integral from ti to tc of R(t) dt minus the integral from ti to tc of F(t) dt, which is equal to:

In equation (9) R(t) is the reaction force at then knob end of the bat, because of the impact, and vcm is the velocity of the center of mass of the slider-bat.  Also if rcm is the distance from the knob to the slider-bat's center of mass, then vcm=rcm*dOc/dt.  It is then presented that the angular impulse-angular momentum equation is equal to the negative integral from ti to tc of rh*F(t) dt, which is equal to the net change in angular momentum, and is equal to the following expression and the negative integral is due to the force F(t) of the ball on the bat is in a direction away from the pitcher:

The author then combines equations (9) and (10) to get the integral from tc to ti of R(t) dt is equal to the reaction impulse and is expressed in the following expression:

The next thing which is done is that the center of percussion is determined, which is the impact point on the slider-bat that corresponds to zero reaction force:

At this point in the article the author has completed most of the numerical analysis that is required to be completed.  The author provides the reader with values for some of the variables for an ordinary bat:

The parameter e is called the coefficient of restitution, which is defined as the negative of the ratio of the relative velocities of the ball and impact point of the slider-bat after and before the impact of the bat and ball.  Also within the article the differential equation which models the swing time until the collision with the bat:

    The end of the article discusses substituting the parameters in for their values in the expressions given, in order to determine their numeric values, and as a result tell the reader whether a slider-bat is effective in a way which is positive for the batter.  The final first order nonlinear initial value problem was solved numerically and this was done using the Euler method, where the derivatives are replaced by their difference quotients.  The article also provides graphical support for their analysis, from which they obtain percentages for the effectiveness of a slider-bat.

History

Since the beginning of baseball, players as well as companies which produce baseball bats have been looking for a way to design their bats so that they perform at a maximum efficiency.  Regulations set by the baseball leagues, however; forbid the use of bats analyzed in the article "Bats and Superbats".  Another method of creating a superbat is to cork a bat; this is where cork is placed in the inside of the bat.  The cork pushes out on the sides of the bat and make the bat exert a much larger force on the ball when hit, than with a normal bat that is passed by the baseball leagues.  This article analyzes whether it is more efficient to use a slider-bat or if it is more efficient to use a normal bat.  The ending result is that if the batter desires a shorter swing time, a normal bat should be used, because the moment of inertia will be smaller than with a slider-bat.  For more power, but a longer swing time, a slider-bat would serve the purpose.  In the history of baseball there has been some players, which have been caught with illegal bats that are meant to serve the same purpose as slider-bats.  It has also been said that a "Superbat" is more in the head of the batter, they do still have to hit the ball whether the bat is a super or non super one.

Conclusion

At the conclusion of the article the reader is presented with an explanation of the effectiveness of a slider-bat.  Through the initial differential equation, which began the analysis, the author determines that the slider is very effective for reducing the swing time, but in return there is a large reduction in the batted-ball velocity.  Some of the energy that the batter puts into the swing is used as kinetic energy into the moving of the slider along its axis, and none of that energy is being transferred to the ball.  The article comes to the conclusion that if the batter desires a shorter swing time, then the batter would be better off to use a normal bat with no slider, which will have a smaller moment of inertia, because the corresponding reduction in va(tf) is lower.  From the authors analysis, a 20% reduction in the swing time, will require a 48% reduction in Ib, which will result in a 10% reduction in the batted-ball velocity.  As a result of this analysis done by the author of the article it was found that the effectiveness of a slider-bat on the batted-ball velocity is not as effective as a normal bat without a slider.

Bibliography

Bailey, Herbert R.  "Bat and Superbat."  The College Mathematics Journal 18:4 (1987):  307-314.